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Improving The Numerical Technique For Computing The Accumulated Distribution Of A Quadratic Form In Normal Variables

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  • Zeng-Hua Lu
  • Maxwell King

Abstract

This paper is concerned with the technique of numerically evaluating the cumulative distribution function of a quadratic form in normal variables. The efficiency of two new truncation bounds and all existing truncation bounds are investigated. We also find that the suggestion in the literature for further splitting truncation errors might reduce computational efficiency, and the optimum splitting rate could be different in different situations. A practical solution is provided. The paper also discusses a modified secant algorithm for finding the critical value of the distribution at any given significance level.

Suggested Citation

  • Zeng-Hua Lu & Maxwell King, 2002. "Improving The Numerical Technique For Computing The Accumulated Distribution Of A Quadratic Form In Normal Variables," Econometric Reviews, Taylor & Francis Journals, vol. 21(2), pages 149-165.
    • Handle: RePEc:taf:emetrv:v:21:y:2002:i:2:p:149-165
    DOI: 10.1081/ETC-120014346
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    References listed on IDEAS

    as
    1. Robert B. Davies, 1980. "The Distribution of a Linear Combination of χ2 Random Variables," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 29(3), pages 323-333, November.
    2. Dufour, Jean-Marie & King, Maxwell L., 1991. "Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or nonstationary AR(1) errors," Journal of Econometrics, Elsevier, vol. 47(1), pages 115-143, January.
    3. Hillier, G.H., 1999. "The density of a quadratic form in a vector uniformly distributed on the n-sphere," Discussion Paper Series In Economics And Econometrics 9902, Economics Division, School of Social Sciences, University of Southampton.
    4. Hillier, Grant, 2001. "THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE," Econometric Theory, Cambridge University Press, vol. 17(1), pages 1-28, February.
    5. Ansley, Craig F. & Kohn, Robert & Shively, Thomas S., 1992. "Computing p-values for the generalized Durbin-Watson and other invariant test statistics," Journal of Econometrics, Elsevier, vol. 54(1-3), pages 277-300.
    6. King, Maxwell L., 1985. "A point optimal test for autoregressive disturbances," Journal of Econometrics, Elsevier, vol. 27(1), pages 21-37, January.
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    Citations

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    Cited by:

    1. Rossi, Francesca & Lieberman, Offer, 2023. "Spatial autoregressions with an extended parameter space and similarity-based weights," Journal of Econometrics, Elsevier, vol. 235(2), pages 1770-1798.
    2. Robinson, Peter M. & Rossi, Francesca, 2015. "Refined Tests For Spatial Correlation," Econometric Theory, Cambridge University Press, vol. 31(6), pages 1249-1280, December.
    3. Lu, Zeng-Hua, 2006. "The numerical evaluation of the probability density function of a quadratic form in normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1986-1996, December.
    4. Begum, Nelufa & King, Maxwell L., 2005. "Most mean powerful test of a composite null against a composite alternative," Computational Statistics & Data Analysis, Elsevier, vol. 49(4), pages 1079-1104, June.
    5. Kan, Raymond & Wang, Xiaolu, 2010. "On the distribution of the sample autocorrelation coefficients," Journal of Econometrics, Elsevier, vol. 154(2), pages 101-121, February.
    6. Lu, Zeng-Hua & King, Maxwell L., 2004. "A Wald-type test of quadratic parametric restrictions," Economics Letters, Elsevier, vol. 83(3), pages 359-364, June.
    7. Aman Ullah & Yong Bao & Yun Wang, 2014. "Exact Distribution of the Mean Reversion Estimator in the Ornstein-Uhlenbeck Process," Working Papers 201413, University of California at Riverside, Department of Economics.

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    More about this item

    Keywords

    Quadratic form in normal variables; Numerical inversion of characteristic function; Truncation error; Newton'; s method; Secant method; JEL Classification ; C19; C63;
    All these keywords.

    JEL classification:

    • C19 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Other
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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